《纺织复合材料》课程参考文献（Textile Composites and Inflatable Structures）07 Wrinkles in Square Membranes

Wrinkles in Square Membranes Y.W.Wong!and S.Pellegrino2 1 SKM Consultants (M)Sdn Bhd,Suite E-15-01,Plaza Mont'Kiara,No.2,Jalan Kiara,Mont'Kiara,50480 Kuala Lumpur,MALAYSIA WYWong@skmconsulting.com.my 2 Department of Engineering,University of Cambridge,Trumpington Street, Cambridge,CB2 1PZ,U.K.pellegrino@eng.cam.ac.uk This paper investigates the wrinkling of square membranes of isotropic mate- rial,subject to coplanar pairs of equal and opposite corner forces.These mem- branes are initially stress free and perfectly flat.Two wrinkling regimes are observed experimentally and are also reproduced by means of finite-element simulations.A general methodology for making preliminary analytical esti- mates of wrinkle patterns and average wrinkle amplitudes and wavelengths, while also gaining physical insight into the wrinkling of membranes,is pre- sented. 1 Introduction Thin,prestressed membranes will be required for the next generation of space- craft,to provide deployable mirror surfaces,solar collectors,sunshields,solar sails,etc.Some applications require membranes that are perfectly smooth in their operational configuration,but many other applications can tolerate membranes that are wrinkled;in such cases the deviation from the nominal shape has to be known.The design of membrane structures with biaxial pre- tension,which would have a smooth surface,significantly increases the overall complexity of the structure and hence,for those applications in which small wrinkles are acceptable,engineers need to be able to estimate the extent, wavelength and amplitude of the wrinkles. The wrinkling of membranes has attracted much interest in the past,start- ing from the development of the tension field theory [1.Simpler formulations and extensions of this theory were later proposed [2-7].All of these formula- tions,with accompanying numerical solutions [8,9,model the membrane as a no-compression,two-dimensional continuum with negligible bending stiffness. Many studies of membrane wrinkling have been carried out during the past three years,and have been presented at the 42nd,43rd,and 44th AIAA SDM Conferences [10-12]. 109 E.Onate and B.Kroplin (eds.).Textile Composites and Inflatable Structures,109-122. 2005 Springer.Printed in the Netherlands

Wrinkles in Square Membranes Y.W. Wong1 and S. Pellegrino2 1 SKM Consultants (M) Sdn Bhd, Suite E-15-01, Plaza Mont’ Kiara, No. 2, Jalan Kiara, Mont’ Kiara, 50480 Kuala Lumpur, MALAYSIA WYWong@skmconsulting.com.my 2 Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, U.K. pellegrino@eng.cam.ac.uk This paper investigates the wrinkling of square membranes of isotropic mate￾rial, subject to coplanar pairs of equal and opposite corner forces. These mem￾branes are initially stress free and perfectly flat. Two wrinkling regimes are observed experimentally and are also reproduced by means of finite-element simulations. A general methodology for making preliminary analytical esti￾mates of wrinkle patterns and average wrinkle amplitudes and wavelengths, while also gaining physical insight into the wrinkling of membranes, is pre￾sented. 1 Introduction Thin, prestressed membranes will be required for the next generation of space￾craft, to provide deployable mirror surfaces, solar collectors, sunshields, solar sails, etc. Some applications require membranes that are perfectly smooth in their operational configuration, but many other applications can tolerate membranes that are wrinkled; in such cases the deviation from the nominal shape has to be known. The design of membrane structures with biaxial pre￾tension, which would have a smooth surface, significantly increases the overall complexity of the structure and hence, for those applications in which small wrinkles are acceptable, engineers need to be able to estimate the extent, wavelength and amplitude of the wrinkles. The wrinkling of membranes has attracted much interest in the past, start￾ing from the development of the tension field theory [1]. Simpler formulations and extensions of this theory were later proposed [2-7]. All of these formula￾tions, with accompanying numerical solutions [8,9], model the membrane as a no-compression, two-dimensional continuum with negligible bending stiffness. Many studies of membrane wrinkling have been carried out during the past three years, and have been presented at the 42nd, 43rd, and 44th AIAA SDM Conferences [10-12]. 109 E. Oñate and B. Kröplin (eds.), Textile Composites and Inflatable Structures, 109–122. © 2005 Springer. Printed in the Netherlands

110 Y.W.Wong and S.Pellegrino This paper considers a uniform elastic square membrane(which is a simple model of a square solar sail)of side length L+2r and thickness t that is prestressed by two pairs of equal and opposite concentrated forces,T1 and T2, uniformly distributed over a small length d at the corners,as shown in Fig.1. This membrane is isotropic with Young's Modulus E and Poisson's ratio v; it is also initially stress free and perfectly flat (before the application of the corner forces). T2.2 T1,δ1 Fig.1.Membrane subjected to corner forces. We use this problem to present a general and yet simple analytical method for making preliminary estimates of wrinkle patterns and average wrinkle am- plitudes and wavelengths in membrane structures.We also present a finite element simulation method for making more accurate estimates.The results from both our analytical approach and finite element simulations are com- pared with experimental measurements. The layout of the paper is as follows.Section 2 describes two regimes of wrinkling that were observed experimentally.Section 3 presents our method- ology for tackling wrinkling problems analytically,and hence derives solutions for the square membrane problem.Section 4 presents a finite-element simula- tion technique,whose results are compared with measurements and analytical results in Section 5.Section 6 concludes the paper. 2 Experimental Observations Fig.2 shows photographs of the wrinkle patterns in a Kapton membrane with L =500 mm,t =0.025 mm,and d =25 mm.For symmetric loading (T1 T2) the wrinkle pattern is fairly symmetric,as shown in Fig.2(a),with wrinkles

110 Y.W. Wong and S. Pellegrino This paper considers a uniform elastic square membrane (which is a simple model of a square solar sail) of side length L + 2r1 and thickness t that is prestressed by two pairs of equal and opposite concentrated forces, T1 and T2, uniformly distributed over a small length d at the corners, as shown in Fig. 1. This membrane is isotropic with Young’s Modulus E and Poisson’s ratio ν; it is also initially stress free and perfectly flat (before the application of the corner forces). Fig. 1. Membrane subjected to corner forces. We use this problem to present a general and yet simple analytical method for making preliminary estimates of wrinkle patterns and average wrinkle am￾plitudes and wavelengths in membrane structures. We also present a finite element simulation method for making more accurate estimates. The results from both our analytical approach and finite element simulations are com￾pared with experimental measurements. The layout of the paper is as follows. Section 2 describes two regimes of wrinkling that were observed experimentally. Section 3 presents our method￾ology for tackling wrinkling problems analytically, and hence derives solutions for the square membrane problem. Section 4 presents a finite-element simula￾tion technique, whose results are compared with measurements and analytical results in Section 5. Section 6 concludes the paper. 2 Experimental Observations Fig. 2 shows photographs of the wrinkle patterns in a Kapton membrane with L = 500 mm, t = 0.025 mm, and d = 25 mm. For symmetric loading (T1 = T2) the wrinkle pattern is fairly symmetric, as shown in Fig. 2(a), with wrinkles T1, δ1 L d T1, δ T2 1 , δ2 T2, δ2 L r1 r1 r1 r1

Wrinkles in Square Membranes 111 radiating from each corner;the central region is free of wrinkles.For a load ratio of T1/T2=2 the wrinkles grow in amplitude but remain concentrated at the corners.Then,for T1/T2=3 a large diagonal wrinkle becomes visible, whose amplitude grows further for T1/T2=4. (a) (b) (c) (d) Fig.2.Wrinkled shapes for Ti equal to (a)5 N,(b)10 N,(c)15 N,and (d)20 N; T2 =5 N in all cases. 3 Analytical Approach Our analytical approach is in four parts,as follows. First,we identify a two-dimensional stress field that involves no compres- sion anywhere in the membrane;the regions where the minor principal stress is zero are then assumed to be wrinkled and the wrinkles are assumed to be along the major principal stress directions.Ideally,both equilibrium and compatibility should be satisfied everywhere by the selected stress field,but

Wrinkles in Square Membranes 111 radiating from each corner; the central region is free of wrinkles. For a load ratio of T1/T2 = 2 the wrinkles grow in amplitude but remain concentrated at the corners. Then, for T1/T2 = 3 a large diagonal wrinkle becomes visible, whose amplitude grows further for T1/T2 = 4. (a) (b) (c) (d) Fig. 2. Wrinkled shapes for T1 equal to (a) 5 N, (b) 10 N, (c) 15 N, and (d) 20 N; T2 = 5 N in all cases. 3 Analytical Approach Our analytical approach is in four parts, as follows. First, we identify a two-dimensional stress field that involves no compres￾sion anywhere in the membrane; the regions where the minor principal stress is zero are then assumed to be wrinkled and the wrinkles are assumed to be along the major principal stress directions. Ideally, both equilibrium and compatibility should be satisfied everywhere by the selected stress field, but

112 Y.W.Wong and S.Pellegrino analytical solutions in closed-form -obtained by tension field theory-ex- ist only for simple boundary conditions.We have recently shown [13 that a carefully chosen,simple stress field that satisfies only equilibrium can provide quick solutions that are useful for preliminary design.More accurate stress fields can be obtained from a two-dimensional stress analysis with membrane finite elements,as briefly discussed in Section 4. Second,we note that the bending stiffness of the membrane is finite,al- though small,and hence a compressive stress will exist in the direction per- pendicular to the wrinkles.Because of its small magnitude,this stress was neglected in our previous analysis of the stress field.We assume that this compressive stress varies only with the wavelength of the wrinkles and set it equal to the critical buckling stress of a thin plate in uniaxial compres- sion.Thus,the stress across the wrinkles is a known function of the wrinkle wavelength. Third,we enforce equilibrium in the out-of-plane direction.Since the stress distribution is known,except for the wrinkle wavelength,a single equation of equilibrium will determine the wrinkle wavelength. Fourth,the wrinkle amplitudes are estimated by considering the total strain in the membrane as the sum of two components,a material strain and a wrinkling strain. 3.1 Stress Field Fig.3 shows three equilibrium stress fields that can be used to analyse membranes under (a)a symmetric loading,(b)an asymmetric loading with Ti/T2≤1/（√2-1），and(c)an asymmetric loading with T/T2≥1/(v2-1): In each case the membrane is divided into regions which are subject to simple stress states. The stress field in Fig.3(a)is purely radial in the corner regions,with T 0r= (1) V2rt where r<r+L/2 is the radial distance measured from the apex.Hence,or is uniform on any circular arc and all other stress components are zero.The central region,defined by circular arcs of radius R=r1+L/2,is subject to uniform biaxial stress of magnitude T/v2Rt. Note that near the point of application of each corner load a small,biaxially stressed region bounded by the radius ri=d/v2 has been defined.In these regions both normal stress components are T/dt. For moderately asymmetric loading,see Fig.3(b),we consider corner stress fields similar to those given by Eq.(1),hence T 0r= (2) V2rt

112 Y.W. Wong and S. Pellegrino analytical solutions in closed-form —obtained by tension field theory— ex￾ist only for simple boundary conditions. We have recently shown [13] that a carefully chosen, simple stress field that satisfies only equilibrium can provide quick solutions that are useful for preliminary design. More accurate stress fields can be obtained from a two-dimensional stress analysis with membrane finite elements, as briefly discussed in Section 4. Second, we note that the bending stiffness of the membrane is finite, al￾though small, and hence a compressive stress will exist in the direction per￾pendicular to the wrinkles. Because of its small magnitude, this stress was neglected in our previous analysis of the stress field. We assume that this compressive stress varies only with the wavelength of the wrinkles and set it equal to the critical buckling stress of a thin plate in uniaxial compres￾sion. Thus, the stress across the wrinkles is a known function of the wrinkle wavelength. Third, we enforce equilibrium in the out-of-plane direction. Since the stress distribution is known, except for the wrinkle wavelength, a single equation of equilibrium will determine the wrinkle wavelength. Fourth, the wrinkle amplitudes are estimated by considering the total strain in the membrane as the sum of two components, a material strain and a wrinkling strain. 3.1 Stress Field Fig. 3 shows three equilibrium stress fields that can be used to analyse membranes under (a) a symmetric loading, (b) an asymmetric loading with T1/T2 ≤ 1/( √2−1), and (c) an asymmetric loading with T1/T2 ≥ 1/( √2−1). In each case the membrane is divided into regions which are subject to simple stress states. The stress field in Fig. 3(a) is purely radial in the corner regions, with σr = T √2rt (1) where r < r1 + L/2 is the radial distance measured from the apex. Hence, σr is uniform on any circular arc and all other stress components are zero. The central region, defined by circular arcs of radius R = r1 + L/2, is subject to uniform biaxial stress of magnitude T /√2Rt. Note that near the point of application of each corner load a small, biaxially stressed region bounded by the radius r1 = d/√ 2 has been defined. In these regions both normal stress components are T/dt. For moderately asymmetric loading, see Fig. 3(b), we consider corner stress fields similar to those given by Eq. (1), hence σr = Ti √2rt (2)

Wrinkles in Square Membranes 113 T2=丁 A R T2-T1 L T=T2 L (a) (b) T 0 201 R 0 L R2 2 0 L (c) Fig.3.Stress fields. but vary the outer radii of these stress fields,in such a way that the radial stress is still uniform on the four arcs bounding the central region.Hence, we need to choose Ri and R2 such that R1/R2 =T1/T2 and R1+R2= L+2r1.This approach is valid until the two larger arcs reach the centre of the membrane,which happens for R1T11 R=五=V2-1 (3) For larger values of T1/T2 we consider the stress field shown in Fig.3(c); note that the diagonal region between the two most heavily loaded corners of the membrane is subject to zero transverse stress,and hence a single di- agonal wrinkle can form.Also note that the edges of the membrane are now unstressed.The stress in each corner region is now given by T 0r= 2rtsini (4)

Wrinkles in Square Membranes 113 Fig. 3. Stress fields. but vary the outer radii of these stress fields, in such a way that the radial stress is still uniform on the four arcs bounding the central region. Hence, we need to choose R1 and R2 such that R1/R2 = T1/T2 and R1 + R2 = L + 2r1. This approach is valid until the two larger arcs reach the centre of the membrane, which happens for R1 R2 = T1 T2 = 1 √2 − 1 (3) For larger values of T1/T2 we consider the stress field shown in Fig. 3(c); note that the diagonal region between the two most heavily loaded corners of the membrane is subject to zero transverse stress, and hence a single di￾agonal wrinkle can form. Also note that the edges of the membrane are now unstressed. The stress in each corner region is now given by σr = Ti 2rtsin θi (4) T2=T1 (a) (b) L L (c) R 0 0 0 0 L T2 T2 T1 T1 T2 T2 T1 T1 T2=T1 T1=T2 R1 R2 R2 R1 T1=T2 2θ1 2θ2 r1 L r1

114 Y.W.Wong and S.Pellegrino and hence for the central region to be biaxially stressed the condition T T2 or=2Ritsin 0 2Rat sin 02 (5) has to be satisfied.Given Ti and T2,one can find-by geometry together with Eq.(5)-a unique set of values for the half-angles defining the corner regions, 01,02,and for the radii,R1,R2,thus fully defining the stress field. The values of 01 and 02 remain constant for any particular value of T1/T2, and so the only variable in Eq.(4)is r.Hence,the slack regions will grow as the load ratio is increased.Finally,it should be noted that both of our earlier stress fields can be obtained as special cases of the last one. 3.2 Wrinkle Details A critical compressive stress,ocr,must exist in the direction transverse to the wrinkles.We will assume that this stress is given by the buckling stress of an infinitely long plate of width A π2E2 0e=-2121-v2) (6) In the case of fan-shaped wrinkles we will set A equal to the half-wavelength mid way between the corner and the edge of the fan. To estimate the wrinkle details,we begin by considering a simple analytical expression for the shape of the wrinkled surface.For example,in the case of a symmetrically loaded membrane we assume that at each corner there is a set of uniform,radial wrinkles whose out-of-plane shape can be described in the polar coordinate system of Fig.4 by w=Asin T(r-ri) sin 2ne Rwrin-T1 (7) where A is the wrinkle amplitude,n the total number of wrinkles at the corner -each subtending an angle of m/2n-and 6 is an angular coordinate measured from the edge of the membrane. Since the stress in the corner regions is uniaxial there is the possibility of wrinkles forming there.The radial strain is Er =Or/E (8) where or is given by Eq.(1).The corresponding radial displacement,u(r) (positive outwards),can be obtained from u=/ erdr+c (9) where the constant of integration c can be obtained by noting that u0 at r=R,i.e.at

114 Y.W. Wong and S. Pellegrino and hence for the central region to be biaxially stressed the condition σr = T1 2R1tsin θ1 = T2 2R2tsin θ2 (5) has to be satisfied. Given T1 and T2, one can find –by geometry together with Eq. (5)– a unique set of values for the half-angles defining the corner regions, θ1, θ2, and for the radii, R1, R2, thus fully defining the stress field. The values of θ1 and θ2 remain constant for any particular value of T1/T2, and so the only variable in Eq. (4) is r. Hence, the slack regions will grow as the load ratio is increased. Finally, it should be noted that both of our earlier stress fields can be obtained as special cases of the last one. 3.2 Wrinkle Details A critical compressive stress, σcr, must exist in the direction transverse to the wrinkles. We will assume that this stress is given by the buckling stress of an infinitely long plate of width λ σcr = −π2 λ2 Et2 12(1 − ν2) (6) In the case of fan-shaped wrinkles we will set λ equal to the half-wavelength mid way between the corner and the edge of the fan. To estimate the wrinkle details, we begin by considering a simple analytical expression for the shape of the wrinkled surface. For example, in the case of a symmetrically loaded membrane we assume that at each corner there is a set of uniform, radial wrinkles whose out-of-plane shape can be described in the polar coordinate system of Fig.4by w = A sin π(r − r1) Rwrin − r1 sin 2nθ (7) where A is the wrinkle amplitude, n the total number of wrinkles at the corner —each subtending an angle of π/2n— and θ is an angular coordinate measured from the edge of the membrane. Since the stress in the corner regions is uniaxial there is the possibility of wrinkles forming there. The radial strain is r = σr/E (8) where σr is given by Eq. (1). The corresponding radial displacement, u(r) (positive outwards), can be obtained from u =  rdr + c (9) where the constant of integration c can be obtained by noting that u ≈ 0 at r = R, i.e. at

Wrinkles in Square Membranes 115 R Rwrin-T1 /2n (b) (a) Fig.4.Corner wrinkles:(a)overall shape;(b)central cross section and definition of half-wavelength. the edge of the biaxially stressed region.Therefore, T u三 V2E血月 (10) The hoop strain required for geometric compatibility is u (11) and the hoop material strain is Or Eom =-V E (12) Wrinkles will form when ee,is larger in magnitude than cem(note that both strains are negative),hence combining Eqs.(1),(10)-(12),we obtain R ln-≥v (13) The radius of the wrinkled region,Rwrin,is the largest r for which Eq.(13) is satisfied.Within the wrinkled region,i.e.for r<Rwrin,an additional "wrin- kling"strain is required E0g E0m +EOwrn (14) The wrinkling strain is related to the wrinkle amplitude,and for the wrinkle shape defined by Eq.(7)it can be shown that at r =(Rwrin-r1)/2 π2A2 E0in=-42 (15)

Wrinkles in Square Membranes 115 Fig. 4. Corner wrinkles: (a) overall shape; (b) central cross section and definition of half-wavelength. the edge of the biaxially stressed region. Therefore, u = T √ 2Et ln r R (10) The hoop strain required for geometric compatibility is θg = u r (11) and the hoop material strain is θm = −ν σr E (12) Wrinkles will form when θg is larger in magnitude than θm (note that both strains are negative), hence combining Eqs. (1), (10)–(12), we obtain ln R r ≥ ν (13) The radius of the wrinkled region, Rwrin, is the largest r for which Eq. (13) is satisfied. Within the wrinkled region, i.e. for r < Rwrin, an additional “wrin￾kling” strain is required θg = θm + θwrin (14) The wrinkling strain is related to the wrinkle amplitude, and for the wrinkle shape defined by Eq. (7) it can be shown that at r = (Rwrin − r1)/2 θwrin = −π2A2 4λ2 (15) π/2n r θ w CL CL CL Rwrin-r1 r1 (b) λ 2A R (a)

116 Y.W.Wong and S.Pellegrino Substituting Eq.(10)into Eq.(11)and Eq.(1)into Eq.(12),and then both into Eq.(14)we find that A has to satisfy v②T V2UT T2A2 、ln Et(Rwrin -T1) Rrin一T1= 2R Et(Rwrin -r1)4X2 (16) Next,we work out the number of wrinkles by considering out-of-plane equilibrium of the wrinkled membrane at a point of maximum out-of-plane displacement,e.g.at r =(Rwrin -r1)/2,0 =/4n.The equilibrium equation 的 Orkr +00K0=0 (17) where Kr and Ke are the curvatures in the radial and hoop directions,respec- tively,which can be obtained by differentiating Eq.(7).Hence, Aπ2 16An2 Kr=一 (Rwrin-r1)2 and o=一(Rwrin-rmP (18) The transverse stress component oo is set equal to oer.Substituting Eqs.(1),(6) and (18)into Eq.(17)gives V27 4Et2n2 (19) (Rwrin-r1)t 3(1-2)2=0 Since A is related to the number of wrinkles by λ= Rwrin -T1 T 2 (20) 2n we can substitute for A into Eq.(19)and solve for n to obtain 3v22T(Rwrin -r1)(1-v2) 64B3 (21) Given Egs.(20)and(21)we can predict the wrinkle amplitude A by solving Eq.(16),which gives 2入 V2T 2R A= (22) Et(Rwrin -T1 Rwrin -r1 In the case T1T2 it is straightforward to generalize Eqs.(19)and (22) to find the wavelength and amplitude of the wrinkles in each corner region. However,for T1/T2 >1/(v2-1)the two larger corner stress fields come into contact,see Fig.3(c),and hence a single diagonal wrinkle can form between the two most heavily loaded corners.This much larger wrinkle can be analysed

116 Y.W. Wong and S. Pellegrino Substituting Eq. (10) into Eq. (11) and Eq. (1) into Eq. (12), and then both into Eq. (14) we find that A has to satisfy √2T Et(Rwrin − r1) ln Rwrin − r1 2R = − √2νT Et(Rwrin − r1) − π2A2 4λ2 (16) Next, we work out the number of wrinkles by considering out-of-plane equilibrium of the wrinkled membrane at a point of maximum out-of-plane displacement, e.g. at r = (Rwrin − r1)/2, θ = π/4n. The equilibrium equation is σrκr + σθκθ = 0 (17) where κr and κθ are the curvatures in the radial and hoop directions, respec￾tively, which can be obtained by differentiating Eq. (7). Hence, κr = − Aπ2 (Rwrin − r1)2 and κθ = − 16An2 (Rwrin − r1)2 (18) The transverse stress component σθ is set equal to σcr. Substituting Eqs. (1), (6) and (18) into Eq. (17) gives √2T (Rwrin − r1)t − 4Et2n2 3(1 − ν2)λ2 = 0 (19) Since λ is related to the number of wrinkles by λ = Rwrin − r1 2 π 2n (20) we can substitute for λ into Eq. (19) and solve for n to obtain n = 4 4 3 √ 2π2T(Rwrin − r1)(1 − ν2) 64Et3 (21) Given Eqs. (20) and (21) we can predict the wrinkle amplitude A by solving Eq. (16), which gives A = 2λ π 4 √2T Et(Rwrin − r1)  ln 2R Rwrin − r1 − ν  (22) In the case T1 = T2 it is straightforward to generalize Eqs. (19) and (22) to find the wavelength and amplitude of the wrinkles in each corner region. However, for T1/T2 ≥ 1/( √2 − 1) the two larger corner stress fields come into contact, see Fig. 3(c), and hence a single diagonal wrinkle can form between the two most heavily loaded corners. This much larger wrinkle can be analysed

Wrinkles in Square Membranes 117 following a similar approach [13],to obtain the following expressions for half- wavelength and amplitude 22R1(R1-r1)2Et3 sin01 (23) 3(1-v2)T1 and A= 2V(d1+d2】 (24) T Here 61 and 62 are the radial displacements of the corners loaded by T and T2,respectively;these corner displacements can be estimated from T d:≈ 2Et sin20; ,h+-(危+-号m i=1,2(25) Here,Ai is the area of a part of the central,biaxially stressed region that is associated with the loads T;.More refined estimates can be obtained from a two-dimensional finite-element analysis,see Section 5. 4 Finite-Element Simulations We have recently shown [14]that wrinkling of a thin membrane can be ac- curately modelled using the thin shell elements available in the commercial finite-element package ABAQUS [15].The analysis is carried out by intro- ducing initial geometrical imperfections,obtained from an initial eigenvalue analysis,followed by a geometrically non-linear post-buckling analysis using the pseudo-dynamic *STABILIZE solution scheme.This approach,although expensive in computational terms,is so far the only method that can reveal full wrinkle details and can be relied upon as an almost exact replication of physical experimentation.An alternative approach is the Iterative Modified Properties (IMP)method [9]which uses a combined stress-strain wrinkling criterion in a two-dimensional membrane model.The IMP method has been recently implemented as an ABAQUS user subroutine and has been shown to accurately predict the extent of the wrinkled regions and the two-dimensional stress distribution-but of course not the details of the wrinkles. Both of these modelling techniques were used to simulate a 0.025 mm thick, 500x 500 mm2 square Kapton membrane(E=3530 N/mm2 and v=0.3).The membrane was loaded at each corner through a spreader beam by a 0.1 mm thick,25 mm x 20 mm Kapton tabs-as in the experiment of Section 2. A uniform mesh of 200 by 200 square elements was used to model the whole structure,in order to capture the fine wrinkle details in the corners. In the shell model,the Kapton membrane and the corner tabs were mod- elled using S4R5 thin shell elements of different thickness.At the corners,the shell elements were connected to "Circ"beam elements through the *MPC

Wrinkles in Square Membranes 117 following a similar approach [13], to obtain the following expressions for half￾wavelength and amplitude λ = 4 4 2π2R1(R1 − r1)2Et3 sin θ1 3(1 − ν2)T1 (23) and A = 2 λ(δ1 + δ2) π (24) Here δ1 and δ2 are the radial displacements of the corners loaded by T1 and T2, respectively; these corner displacements can be estimated from δi ≈ Ti 2Etsin2 θi θi ln Ri r1 + (1 − ν)  Ai R2 i + θi − 1 2 tan θi  i = 1, 2 (25) Here, Ai is the area of a part of the central, biaxially stressed region that is associated with the loads Ti. More refined estimates can be obtained from a two-dimensional finite-element analysis, see Section 5. 4 Finite-Element Simulations We have recently shown [14] that wrinkling of a thin membrane can be ac￾curately modelled using the thin shell elements available in the commercial finite-element package ABAQUS [15]. The analysis is carried out by intro￾ducing initial geometrical imperfections, obtained from an initial eigenvalue analysis, followed by a geometrically non-linear post-buckling analysis using the pseudo-dynamic *STABILIZE solution scheme. This approach, although expensive in computational terms, is so far the only method that can reveal full wrinkle details and can be relied upon as an almost exact replication of physical experimentation. An alternative approach is the Iterative Modified Properties (IMP) method [9] which uses a combined stress-strain wrinkling criterion in a two-dimensional membrane model. The IMP method has been recently implemented as an ABAQUS user subroutine and has been shown to accurately predict the extent of the wrinkled regions and the two-dimensional stress distribution —but of course not the details of the wrinkles. Both of these modelling techniques were used to simulate a 0.025 mm thick, 500×500 mm2 square Kapton membrane (E=3530 N/mm2 and ν = 0.3). The membrane was loaded at each corner through a spreader beam by a 0.1 mm thick, 25 mm × 20 mm Kapton tabs—as in the experiment of Section 2. A uniform mesh of 200 by 200 square elements was used to model the whole structure, in order to capture the fine wrinkle details in the corners. In the shell model, the Kapton membrane and the corner tabs were mod￾elled using S4R5 thin shell elements of different thickness. At the corners, the shell elements were connected to “Circ” beam elements through the *MPC

118 Y.W.Wong and S.Pellegrino TIE function.The central node was constrained against translation in the z- and y-direction;all side edges were left free.Both the out-of-plane rotations of the membrane and all in-plane bending degrees of freedom of the corner beams were restrained.The corner loads were applied as distributed loads along the truncated corners of the membrane. In the IMP model,the membrane was modelled using M3D4 membrane elements,whose constitutive behaviour was modelled through a UMAT sub- routine.The corner tabs were modelled with S4 shell elements and the same beam elements as for the shell model were used. 4.1 Simulation Details Two load steps were applied,first a symmetric loading of T=T2=5 N. Second,T2=5 N was maintained while T was increased up to 20 N. The analysis procedure was essentially identical for all of the simulations. First,a uniform prestress of 0.5 N/mm2 was applied,to provide initial out- of-plane stiffness to the membrane.This was achieved by means of *INITIAL CONDITION,TYPE=STRESS.Next,a non-linear-geometry analysis was carried out,with *NLGEOM,to check the equilibrium of the prestressed sys- tem.Then,a linear eigenvalue analysis step was carried out(for the thin shell model only)in order to extract possible wrinkling mode-shapes of the mem- brane under a symmetrical loading.Four such mode-shapes were selected, based on their resemblance to the expected final wrinkled shape,and were introduced as initial geometrical imperfections.Finally,an automatically sta- bilised post-wrinkling analysis was performed,with *STATIC,STABILIZE. This analysis is very sensitive because the magnitude of the wrinkles is very small,hence an increment of 0.001 of the total load had to be selected.The de- fault stabilize factor was reduced to 10-12 to minimise the amount of fictitious damping;this was the smallest amount required to stabilise the solution. Fig.5(a)shows the symmetrically wrinkled shape obtained for T1=T2= 5 N.The wrinkle amplitudes are very small and have been enlarged 100 times for clarity.Fig.5(b)shows the corresponding shape for T1/T2=4.Here the distinguishing feature is a large diagonal wrinkle between the two more heavily loaded corners,a number of smaller wrinkles can also be seen near the corners with smaller loads. Fig.6 shows the wrinkle profiles measured at three different cross sections, for T1/T2=4,plotted against those obtained from the simulations.Note that experiments and simulations match closely in the central region;the wrinkle wavelengths,in particular,are predicted quite accurately.But ABAQUS pre dicts smaller displacements of the edges of the membrane,due to the fact that the initial shape of the physical model has not been captured with sufficient accuracy (the edges of a Kapton sheet are naturally curled)

118 Y.W. Wong and S. Pellegrino TIE function. The central node was constrained against translation in the x￾and y-direction; all side edges were left free. Both the out-of-plane rotations of the membrane and all in-plane bending degrees of freedom of the corner beams were restrained. The corner loads were applied as distributed loads along the truncated corners of the membrane. In the IMP model, the membrane was modelled using M3D4 membrane elements, whose constitutive behaviour was modelled through a UMAT sub￾routine. The corner tabs were modelled with S4 shell elements and the same beam elements as for the shell model were used. 4.1 Simulation Details Two load steps were applied, first a symmetric loading of T1 = T2 = 5 N. Second, T2 = 5 N was maintained while T1 was increased up to 20 N. The analysis procedure was essentially identical for all of the simulations. First, a uniform prestress of 0.5 N/mm2 was applied, to provide initial out￾of-plane stiffness to the membrane. This was achieved by means of *INITIAL CONDITION, TYPE=STRESS. Next, a non-linear-geometry analysis was carried out, with *NLGEOM, to check the equilibrium of the prestressed sys￾tem. Then, a linear eigenvalue analysis step was carried out (for the thin shell model only) in order to extract possible wrinkling mode-shapes of the mem￾brane under a symmetrical loading. Four such mode-shapes were selected, based on their resemblance to the expected final wrinkled shape, and were introduced as initial geometrical imperfections. Finally, an automatically sta￾bilised post-wrinkling analysis was performed, with *STATIC, STABILIZE. This analysis is very sensitive because the magnitude of the wrinkles is very small, hence an increment of 0.001 of the total load had to be selected. The de￾fault stabilize factor was reduced to 10−12 to minimise the amount of fictitious damping; this was the smallest amount required to stabilise the solution. Fig. 5(a) shows the symmetrically wrinkled shape obtained for T1 = T2 = 5 N. The wrinkle amplitudes are very small and have been enlarged 100 times for clarity. Fig. 5(b) shows the corresponding shape for T1/T2 = 4. Here the distinguishing feature is a large diagonal wrinkle between the two more heavily loaded corners, a number of smaller wrinkles can also be seen near the corners with smaller loads. Fig. 6 shows the wrinkle profiles measured at three different cross sections, for T1/T2 = 4, plotted against those obtained from the simulations. Note that experiments and simulations match closely in the central region; the wrinkle wavelengths, in particular, are predicted quite accurately. But ABAQUS pre￾dicts smaller displacements of the edges of the membrane, due to the fact that the initial shape of the physical model has not been captured with sufficient accuracy (the edges of a Kapton sheet are naturally curled)